A counterexample to the dominating set conjecture

The metric polytope met _n is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities x _ij − x _ik − x _jk ≤ 0 and x _ij + x _ik + x _jk ≤ 2 for all triples i, j, k of {1,..., n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n ≤ 8 and, in particular, for the 1,550,825,600 vertices of met₈. While the overwhelming majority of the known vertices of met₉ satisfy the conjecture, we exhibit a fractional vertex not adjacent to any integral vertex.     

A conjecture of Erdös on 3-powerful numbers

Erdös conjectured that the Diophantine equation has infinitely many solutions in pairwise coprime 3-powerful integers, i.e., positive integers for which implies . This was recently proved by Nitaj who, however, was unable to verify the further conjecture that this could be done infinitely often with integers , and none of which is a perfect cube. This is now demonstrated.

     

More on the duality conjecture for entropy numbers

We verify, up to a logarithmic factor, the duality conjecture for entropy numbers in the case where one of the bodies is an ellipsoid. To cite this article: S. Artstein et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A generalized version of Branner-Hubbard conjecture for rational functions

In 1992, Branner and Hubbard raised a conjecture that the Julia set of a polynomial is a Cantor set if and only if each critical component of its filled-in Julia set is not periodic. This conjecture was solved recently. In this paper, we generalize this result to a class of rational functions.     

On a conjecture of the Euler numbers

The main purpose of this paper is to prove a conjecture of the Euler numbers and its generalization by using the analytic methods. That is, for any prime and integer α?1 we proved , where E2n are the Euler numbers and ?(n) the Euler function.     

A conjecture of recski in combinatorial set theory

A p-cover of $\text{n = {1, 2,…,}\text{n}\text{}}$ is a family of subsets $\text{S}{}_{\mathrm{i}}\text{≠ ?}$ such that ∪ S_i = n and |S_i ∩ S_i| ? p for i ≠ j. We prove that for fixed p, the number of p-cover of n is O(n^p+1logn).     

Proof of a conjecture on fractional Ramsey numbers

Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function r_f (a₁, a₂, …, a_k) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k=2. In this article, we answer an open problem by determining an explicit formula for the general case k>2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 164–178, 2010     

关于有理数幂分数部分的注记

熟知有理数幂分数部分的研究与华林问题有密切关系。1957年Mahler证明了,对任意互素整数a,q满足a>q>2和任意ε>0,只有有限多个整数n使得 ‖(a/q)~n‖k,不等式     

The Atiyah-Jones conjecture for rational surfaces

We show that if the Atiyah-Jones conjecture holds for a surface X, then it also holds for the blow-up of X at a point. Since the conjecture is known to hold for P² and for ruled surfaces, it follows that the conjecture is true for all rational surfaces.     

The dual spectral set conjecture

Suppose that where are real numbers such that and The union is not assumed to be disjoint. It is shown that the translates , , tile the real line for some bounded measurable set if and only if the exponentials , , form an orthogonal basis for some bounded measurable set

     

Which rational numbers are binding numbers?

The concept of the binding number of a graph was introduced by Woodall in 1973. in this paper we characterize the set F_n of all pairs (a, b) of integers such that there is a graph G with n vertices and binding number a/b that has a realizing set of b vertices.     

Normalized difference set tiling conjecture

A difference set tiling in a group G is a collection of its difference sets that partition . It can exist in an abelian as well as in a nonabelian group. A tiling is normalized if a product of elements in each difference set equals 1. All known cases in abelian groups are normalized. Ćustić, Krčadinac, and Zhou made a conjecture that this is necessary. We will call it a normalized tiling conjecture (NTC). Using character theory, we prove that NTC is true for where v is odd. Also, if difference set has a multiplier, we prove that NTC is also true.     

Proof of a conjecture on multisets of hook numbers

A conjecture of Regev and Vishik on the equality of two multisets of hook numbers is proved. Supported in part by N.S.F. Grant No. DMS-94-01197. Supported in part by N.S.F. Grant No. DMS-95-00646.     

A method for exact computation with rational numbers

An easily programmed method is presented for solving N linear equations in N unknowns exactly for the rational answers, given that all coefficients and constants appearing in the equations are rational numbers. The rational answers are deduced from floating point approximations to the answers obtained by any of the standard solution algorithms. Criteria are given for determining for a particular set of equations the floating point precision needed.     

On the product sets of rational numbers

A new lower bound on the size of product sets of rational numbers is obtained. An upper estimate for the multiplicative energy of two sets of rational numbers is also found.     

A coloured window on pre-service teachers’ conceptions of rational numbers

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities to construct understandings and representations than initial learners, some of which may be more complex and engrained; pre-service teachers are likely to have created deeply-held–and often negative–beliefs and attitudes toward certain mathematical ideas and processes. In our recent research, we found that pre-service teachers who used a particular computer-based microworld, one emphasising visual representations of and experimental interactions with elementary number theory concepts, overcame many cognitive and psychological difficulties reported in the literature. In this study, we investigate the possibilities of using a similarly-designed microworld that involves a set of rational number concepts. We describe the affordances of this microworld, both in terms of pre-service teacher learning and research on pre-service teacher learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-service teachers. We also show how their interactions with this microworld provided many with a new and aesthetically-rich set of visualisations and experiences.